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Chapter 3: Problem 36

Find the maximum or minimum value of the function. $$f(t)=10 t^{2}+40 t+113$$

Short Answer

Expert verified
The minimum value of the function is 73.

Step by step solution

01

Identify the Quadratic Function

The function given is a quadratic function in the form of \( f(t) = at^2 + bt + c \), where \( a = 10 \), \( b = 40 \), and \( c = 113 \). In this case, \( a \) is positive, which means the parabola opens upwards, indicating a minimum value point at its vertex.
02

Find the Vertex of the Parabola

Use the vertex formula for a quadratic function, which is \( t = \frac{-b}{2a} \). Substitute \( a = 10 \) and \( b = 40 \) into the formula to find the \( t \)-coordinate of the vertex. This results in \( t = \frac{-40}{2(10)} = -2 \).
03

Calculate the Function Value at the Vertex

Substitute \( t = -2 \) back into the original function to find \( f(-2) \). Compute \( f(-2) = 10(-2)^2 + 40(-2) + 113 = 40 - 80 + 113 = 73 \).
04

Determine the Minimum Value

Since the parabola opens upwards and the vertex \( t = -2 \) gives \( f(t) = 73 \), the minimum value of the function is \( 73 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vertex Formula
The vertex formula is a crucial tool for working with quadratic functions. It helps us find the vertex, which is the highest or lowest point on a parabola, depending on whether it opens upwards or downwards. For any quadratic function in the standard form \( f(t) = at^2 + bt + c \):
  • The vertex \( t \)-coordinate can be found using the formula \( t = \frac{-b}{2a} \).
  • This formula derives from the process of completing the square and highlights the symmetrical nature of parabolas around their vertex.
      • Substituting the coefficients \( a \) and \( b \) from our quadratic equation, you can efficiently find this \( t \)-value, allowing further evaluation or graphing of the quadratic.
Parabola
A parabola is a curved, U-shaped graph formed by plotting a quadratic equation. Parabolas appear naturally in many real-world scenarios, such as the path of projectiles. The shape of the parabola provides important features:
  • It has an axis of symmetry, which runs through the vertex.
  • The direction in which the parabola opens is determined by the sign of the \( a \)-coefficient: positive opens upwards, negative opens downwards.
  • The vertex of the parabola is the turning point, either a maximum or minimum value.
Understanding these properties helps in predicting how the graph will behave and how the function's values change as \( t \) varies.
Minimum Value
In a quadratic function where the parabola opens upwards (which happens if \( a > 0 \)), the vertex represents the minimum point. To find the minimum value of the function:
  • Use the vertex formula to obtain the \( t \)-coordinate of the vertex.
  • Substitute this \( t \)-value back into the original quadratic equation.
  • The resulting \( f(t) \) value is the minimum value of the function.
In the given exercise, substituting \( t = -2 \) into the function yielded \( f(t) = 73 \). This means 73 is the minimum value of the function, confirming that the vertex delivers the lowest point on the parabola.
Quadratic Equations
Quadratic equations are polynomials of degree two, having the general form \( ax^2 + bx + c = 0 \). The characteristics of quadratic equations include:
  • A parabolic graph, which can be identified readily due to its distinctive U-shape.
  • The potential to have zero, one, or two real roots based upon the discriminant \( b^2 - 4ac \).
  • Always includes a maximum or minimum value at its vertex.
  • Can be solved using methods like factoring, completing the square, or the quadratic formula.
Understanding the structure and solving methods for quadratic equations forms the foundation for exploring more complex algebraic concepts.

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